Recall for subspaces U,V of R",U + V = {u+v: u € U, v e V}. This is asubspace of R" referred to as the sum of U and V.In this problem you are asked to prove thatdim(U + V) = dim(U)+ dim(V) – dim(U n V).Hint: You need to show that U + Ww has a basis with sizedim(U) + dim(W) – dim(U n W).Recall a basis is of a subspace X of R" is a sequence of vectors from X whichis linearly independent and spans X.

The proof of given problem is given as follows :

Recall for subspaces U, V of R",U +V = {u+v : u e U, v e V}. This is a subspace of R" referred to as the sum of U and V. In this problem you are asked to prove that dim(U + V) = dim(U) + dim(V) – dim(Un V). Hint: You need to show that U + W has a basis with size dim(U) + dim(W) – dim(U n W). Recall a basis is of a subspace X of R" is a sequence of vectors from X which is linearly independent and spans X.

Recall for subspaces U, V of R",U +V = {u+v : u e U, v e V}. This is a subspace of R" referred to as the sum of U and V. In this problem you are asked to prove that dim(U + V) = dim(U) + dim(V) – dim(Un V). Hint: You need to show that U + W has a basis with size dim(U) + dim(W) – dim(U n W). Recall a basis is of a subspace X of R" is a sequence of vectors from X which is linearly independent and spans X.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Recall for subspaces U,V of R",U + V = {u+v: u € U, v e V}. This is a
subspace of R" referred to as the sum of U and V.
In this problem you are asked to prove that
dim(U + V) = dim(U)+ dim(V) – dim(U n V).
Hint: You need to show that U + Ww has a basis with size
dim(U) + dim(W) – dim(U n W).
Recall a basis is of a subspace X of R" is a sequence of vectors from X which
is linearly independent and spans X.

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